Calculating Acceleration of Stellar Engines: A Comparison to Wikipedia Example

[mistergrinch]

Has anyone here looked at the physics of stellar engines? See http://en.wikipedia.org/wiki/Stellar_engine for an overview. In particular, I have done a simple calculation for the acceleration due to light pressure of a "Shkadov thruster", which is a reflective shell on one side of a star that stays in place due to the equilibrium of gravity and light pressure. My calculation for the acceleration this would produce on a star seems to be off by a factor of 4 from the Wikipedia example, which states:

For a star such as the Sun, with luminosity 3.85 × 10^26 W and mass 1.99 × 10^30 kg, the total thrust produced by reflecting half of the solar output would be 1.28 × 10^18 N. After a period of one million years this would yield an imparted speed of 20 m/s, with a displacement from the original position of 0.03 light-years. After one billion years, the speed would be 20 km/s and the displacement 34,000 light-years

Is anyone able to derive these numbers? Thanks!

 

[qraal]

I got the same as the Wikipedia quote. How did you derive what you got?

 

[mistergrinch]

Hmm OK I guess I don't understand the physics then. I'm thinking of photons being emitted in all directions, but half of them are hitting the reflective shell, which transfers momentum to the shell which then tugs on the star by gravity. I assumed an absorbing shell, because it seems to me that a reflective shell would just bounce the photons back to the star, counteracting the gravity tug and the star would go nowhere.

I used the radiation pressure P = Intensity/c = L /(4pi*r^2*c), where L is the luminosity of the sun.

If I calculate the radiation force on the shell in the x direction using differential ring elements I get the equation

dF = P*dA = L/(4*pi*r^2*c) * 2*pi*r^2 * sinB*dB =>
dF_x = L/2c * sinB * cosB dB

where B is the angle from the axis of the ring to the edge

Integrating this from A=0 to pi/2 gives F_x = L/4c = 3.25 * 10^17, which is 1/4 the value given in Wikipedia. Where am I going wrong here?

 

[qraal]

Hi mistergrinch

First, the Skhadov thruster is a half-shell. Half the Sun is exposed, thus the emission from that direction produces net thrust too, because light directed at the shell is bounced rearwards. The shell is shaped like a concentrating mirror, but working in reverse so the light emitted at the the focus, by the star, is reflected away from the mirror (and star) to produce thrust. The Wikipedia diagram doesn't seem quite right. The mirror needs to cover more solid angle in the direction of motion to produce the most thrust. My thrust figure is an idealization, but should be pretty close. The Thruster would need to be made of super-materials to be strong enough at that scale.

 

[mistergrinch]

Oh I get it now, you can shape the shell any way you want if you choose the shell density so that radiation pressure cancels gravity at any distance. So you shape the shell like a paraboloid with the sun at the focus. If you crunch the numbers for a reflective shell you get F = L/c. Thanks!

 

[qraal]

Oh I get it now, you can shape the shell any way you want if you choose the shell density so that radiation pressure cancels gravity at any distance. So you shape the shell like a paraboloid with the sun at the focus. If you crunch the numbers for a reflective shell you get F = L/c. Thanks!

Right on.